Formal Definition
Another way to describe the localization of a ring R at a subset S is via category theory. If R is a ring and S is a subset, consider all R-algebras A, so that, under the canonical homomorphism R → A, every element of S is mapped to a unit. These algebras are the objects of a category, with R-algebra homomorphisms as morphisms. Then, the localization of R at S is the initial object of this category.
Read more about this topic: Localization Of A Ring
Famous quotes containing the words formal and/or definition:
“Two clergymen disputing whether ordination would be valid without the imposition of both hands, the more formal one said, “Do you think the Holy Dove could fly down with only one wing?””
—Horace Walpole (1717–1797)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)